O objetivo principal desta tese e a construção do semigrupo inverso das classes de isomorfismo das extensões abelianas parciais de um anel comutativo. / The main purpose of this thesis is the construction of the inverse semigroup of the isomorphism classes of the partial abelian extensions of a commutative ring.

O objetivo deste trabalho é mostrar que o conjunto de todas G-super palavras monótonas restritas escritas com super letras duras forma uma base para a álgebra de Hopf H, onde H é gerada por um conjunto skew-primitivo semi-invariante {a1, ..., an} e um grupo abeliano G de todos elementos grouplike. / The objective of this work is to show that the set of all monotonic restricted G-super-words written with hard super-letters form a basis for the Hopf algebra H, where H is generated by a skew-primitive semi-invariant set {a1, ..., an} and an abelian group G of all group-like elements.

Neste trabalho discutimos a semissimplicidade de álgebras de Hopf finito-dimensionais e construímos o Duplo de Drinfeld D(H) de uma tal álgebra H. Além disso, apresentamos um resultado mostrando a equivalência entre as categorias de representações dos módulos sobre D(H) e dos módulos de Yetter-Drinfeld sobre Hcop. Como consequência deste estudo, apresentamos um resultado que caracteriza uma álgebra de Hopf quase triangular. / In this work we discuss the semisimplicity of some finite-dimensional Hopf Algebras and we set up the Drinfel’d double D(H) of such an algebra H. In addiction, we present a result showing the equivalence between the representation category of modules over D(H) and the Yetter-Drinfeld modules over Hcop. As a consequence of this, we present a result that features a quasitriangular Hopf algebra.

O objetivo deste trabalho é mostrar que o conjunto de todas G-super palavras monótonas restritas escritas com super letras duras forma uma base para a álgebra de Hopf H, onde H é gerada por um conjunto skew-primitivo semi-invariante {a1, ..., an} e um grupo abeliano G de todos elementos grouplike. / The objective of this work is to show that the set of all monotonic restricted G-super-words written with hard super-letters form a basis for the Hopf algebra H, where H is generated by a skew-primitive semi-invariant set {a1, ..., an} and an abelian group G of all group-like elements.

In this thesis we show that it is impossible to define a Hopf multiplication on the wedge X V Y of a connected finite CW-complex X with an arbitrary connected CW-complex Y, provided neither X nor Y homotopy equivalent to a point. We also show by an example, that the above is no longer true, if one does not require that X is a finite CW-complex. / Science, Faculty of / Mathematics, Department of / Graduate

Notre travail s’inscrit dans un domaine initié en 1934 par Lusternik et Schnirelmann qui associent a une variété un invariant appelé catégorie qui permet de minorer le nombre des points critiques d’une fonction différentiable sur cette variété. Nous nous intéressons à une généralisation au cas des applications continues entre espaces topologiques, auxquelles nous associons un invariant appelé sigma-i-catégorie. Nous obtenons plusieurs caractérisations de la sigma-i-catégorie d’une application. Nous examinons ensuite l’effet sur la sigma-i-catégorie d’un attachement d’une cellule à la source d’une application. Cette étude est faite au moyen d’un nouvel invariant, appelé invariant de Hopf relatif. Enfin nous examinons les relations entre les catégories de produit et de produit smash. / Our work is registered in a field initiated in 1934 by Lusternik and Schnirelmann, which associate with a variety an invariant called category, which allows to undervalue the number of the critical points of a differentiable function on this variety. We are interested in a generalization in the case of the continuous applications between topological spaces in wich we associate an invariant called sigma-i-category. We obtain several characterizations of the sigma-i-category on an application We examine then the effect on the sigma-i-category of a cell attachment on an application source. This study is made with a new invariant, called invariant of relative Hopf. Finally we examine the relations between the categories of product and product smach.

Neste trabalho discutimos a semissimplicidade de álgebras de Hopf finito-dimensionais e construímos o Duplo de Drinfeld D(H) de uma tal álgebra H. Além disso, apresentamos um resultado mostrando a equivalência entre as categorias de representações dos módulos sobre D(H) e dos módulos de Yetter-Drinfeld sobre Hcop. Como consequência deste estudo, apresentamos um resultado que caracteriza uma álgebra de Hopf quase triangular. / In this work we discuss the semisimplicity of some finite-dimensional Hopf Algebras and we set up the Drinfel’d double D(H) of such an algebra H. In addiction, we present a result showing the equivalence between the representation category of modules over D(H) and the Yetter-Drinfeld modules over Hcop. As a consequence of this, we present a result that features a quasitriangular Hopf algebra.

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